Research On The Game Problem Of Cooperative Investment Strategy Selection Between Investors In Continuous Time

The problem of optimal portfolio selection always be the core of portfolio theory research.It studies the optimal allocation of assets in uncertain financial environments,which achieve an equilibrium between maximizing returns and minimizing risk under different objective functions.Based on a reasonable allocation of limited wealth,it is difficult for an investor to obtain a portfolio that has a good risk diversification and does not trigger higher transaction costs.Therefore,consider investor grouping.Reducing the impact of constraints such as a lack of capital,insufficient experience,and different fields on individual investors through mutual cooperation,so as to achieve investment specialization,investment diversification,and lower transaction costs.It’s truly feasible that win-win cooperation is existing in actual transactions.In this paper,under the goal of maximizing the expected terminal wealth utility,we systematically studies the cooperative investment strategy selection game problem between investors with exponential and power utility functions,and explores the conditions when investors can establish a stable cooperation.This is of great significance to improve portfolio theory and guide actual investment activities.First,we consider the cooperative choice game between two investors under the objective function of maximizing expected utility.Assuming that the return of two risky stocks in the incomplete financial market follows normal distribution and has positively correlated.Assuming all participants meet the exponential utility function,using the principle of dynamic programming to obtain the Nash equilibrium investment strategy and value function of investors in two special and general cases.The result shows that the redistribution of investors’risk bearing in the cooperative environment depends on the risk aversion coefficient of investors and the correlation parameters of stocks.When the returns of stocks are independent,cooperation will increase the risk bearing capacity of investors,and when there is a strong positive correlation between two risky stocks,cooperation can lead to investors reducing their risk taking.The simultaneously positive promotion in two objective functions is the core to maintain the cooperative relationship.In the special case where the stock returns are independent,at the end of the period,the premise that participants in the cooperation can obtain the utility benefits are depend on the ratio of the Sharp ratio of risk assets and the relative risk aversion coefficient.In general,if all investors want to have the increment of terminal utility at the same time,they need to meet stricter limits,so as the range of the feasible relative risk aversion coefficient will also be narrower.Secondly,considering the cooperative choice game problem of investors in different utility functions,under the framework of expected utility maximization,the Nash equilibrium investment strategy with power utility function and the corresponding objective function are obtained.Through numerical calculation and sensitivity analysis,we discuss the relationship between Nash equilibrium investment strategy,the value function and the main parameters of the model,obtain the conditions where investors can have stable cooperations.The research shows investors’risk taking in cooperation also depends on the change of total wealth level.Moreover,the conditions that can change the risk bearing capacity under special circumstances are also affected by the proportion of wealth distribution and the way of wealth distribution.If two risk stocks are independent from each other,when the ratio of investors’relative risk aversion coefficient meets0.5（27）p₁/p₂（27）2,both investors participating in the cooperation can obtain the final utility income.At this time,the cooperation can achieve win-win and the cooperation relationship is stable.When the return rates of risky stocks meet the general conditions,the p₁/p₂ feasible range obtained under the power utility function is narrower than that obtained under the exponential utility function,and the cooperation between the two parties also needs to meet more stringent conditions.